Oscillator Duffing

Figure 12. The dependence of the activation energy R on the amplitude A of the harmonic driving force F(t) =A cos (1.2r) as determined [141] by electronic experiment (filled circles), numerical sumulations (open circles) and analytical calculation (solid line), based on (28) for an overdamped duffing oscillator U q) = —q2/2 +

Let us now formulate the problem of the energy-optimal steering of the motion from a chaotic attractor to the coexisting stable limit cycle for a simple model, a noncentrosymmetric Duffing oscillator. This is the model that, in the absence of fluctuations, has traditionally been considered in connection with a variety of problems in nonlinear optics [166]. Consider the motion of a periodically driven nonlinear oscillator under control... 

Find an approximate relation between the amplitude and frequency of the Duffing oscillator x -4- x -t- % = 0, where can have either sign. Interpret the results physically. 

Two comments (1) This exercise shows that the Duffing oscillator has a frequency that depends on amplitude w = 1-i- a -I-O( ), in agreement with (7.6.57). (2) The Poincare-Lindstedt method is good for approximating periodic solutions, but that s all it can do if you want to explore transients or nonperiodic solutions, you can t use this method. Use two-timing or averaging theory instead. 

Exercises 8.4.5-8.4.11 deal with the forced Duffing oscillator in the limit where the forcing, detuning, damping, and nonlinearity are all weak ... 

D. Permann and I. Hamilton, Wavelet Analysis of Time Series for the Duffing Oscillator The Detection of Order Within Chaos, Physical Review Letters, 69 (1992), 2607-2610. 

Keywords ambient vibration correlation function Duffing oscillator hydraulic jump information entropy modal identification optimal sensor placement spectral density structural health monitoring Wishart distribution... 

The first application is concerned with the nonlinear Duffing oscillator of known mass M = 1.0 kg subjected to zero-mean stationary Gaussian white noise / with spectral intensity Sfo. ... 

By solving Equations (3.81)-(3.83), the response variance of the Duffing oscillator can be approximated by ... 

This case is, again, unidentifiable. The updated PDF is plotted together with the previous one in Figure 3.16 and the trajectories of the peaks in the (A i, K3) plane have different slopes. By Equation (3.79), the equivaient linear system has a stiffness Ki + 3a K3 so different Duffing oscillators with K - - = K (a constant) are associated with the same equivalent linear... 

Table 3.4 Comparison of the actual parameters versus the optimal estimates and their statistics for the Duffing oscillator...

Figure 3.20 Response spectrum of the Duffing oscillator with S/o = 0.08 s...

If this expected photoemission really takes place, the resultant spectra should reflect the nonhnear dynamics of nonadiabatic vibrational motion under an external field, which is similar to classical driven oscillators such as a forced Duffing oscillator [156, 239]. Therefore various nonlinear phenomena such as limit cycle, frequency locking, and chaos (1.5-dimensional chaos) [156, 239] can be expected, which would be intrinsically originated from the quantmn dynamics. Furthermore, one may be able to control the frequency and amplitude of the photoemission by varying the laser parameters applied. It may be possible to utilize the photoemission as a new optical somce and also as finger-print signals to identify molecular species and/or molecular states. In this section we illustrate the appearance of such... 

The differences and capabilities of the methods described above are shown by an application to two different nonlinear systems. The first system under investigation Is the slngle-degree-of-freedom Duffing oscillator. Its motion Is defined by... 

Fig. 4 Exceedance Probability of the Response of a SDOF Duffing Oscillator...

Noise-induced transitions have been studied theoretically in quite a few physical and chemical systems, namely the optical bistability [12,13,5], the Freedricksz transition in nematics [14,15,16,5], the superfluid turbulence in helium II [17], the dye laser [18,19], in photochemical reactions [20], the van der Pol-Duffing oscillator [21] and other nonlinear oscillators [22]. Here I will present a very simple model which exhibits a noise-induced critical point. The so-called genetic model was first discussed in [4]. I will not describe its application to population genetics in this paper, see [5] for this aspect, but use a chemical model reaction scheme ... 

The developed algorithm is used to obtain the response of a few randomly excited dynamical systems. A Duffing oscillator is considered whose... 

The problem of estimating the response of a randomly excited Duffing oscillator is considered. The governing equation of motion of a Duffing oscillator is given by... 

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